How to count nth percentile from normally distributed random variable?I have normally distributed random variable X ∼ N ( 100 , 225 ). How to count nth percentile?In my case I need lower quartile - x ( 0.25 )

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How to count   nth percentile from normally distributed random variable?I have normally distributed random variable   X  ∼      N    (  100  ,  225  ). How to count   nth percentile?In my case I need lower quartile -   x  (  0.25  )

upornompe · Accepted Answer

In general, if   X  ∼      N    (  μ  ,      σ    2    ), then            X      −      μ        σ    ∼      N    (  0  ,  1  )follows a standard normal distribution. If       F    X   denotes the cumulative distribution function of   X and   Φ is that of a standard normal distribution, then      F    X    (  x  )  =  P  (  X  ≤  x  )  =  Φ      (                  x        −        μ            σ        )    .Let us find the   pth quantile/percentile of   X, where   p  ∈  (  0  ,  1  ), in terms of the   pth quantile/percentile of a standard normal distribution. Let   y denote the   pth quantile of a standard normal distribution, i.e.  Φ  (      y    p    )  =  p  .If we let       x    p    =  σ      y    p    +  μ then      F    X    (      x    p    )  =  Φ      (                            x          p                −        μ            σ        )    =  Φ  (      y    p    )  =  p  ,i.e.        x    p   is the   pth percent of   XSo all you need to know is the   pth quantile of the standard normal distribution which can be found in tables, and then you can find   pth quantiles for every normal distribution.Note that some tables only index probabilities   p  ≥  0.5, because the rest can be deduced from these quantiles by symmetry. Recall that   1  −  Φ  (  x  )  =  Φ  (  −  x  ) for all   x. If   q  &amp;lt;  0.5 and   p  =  1  −  q, then   p  ≥  0.5 and we can look up the   pth quantile       x    p  . By symmetry:  q  =  1  −  p  =  1  −  Φ  (      x    p    )  =  Φ  (  −      x    p    )and so       x    q    =  −      x    p  , i.e. the   qth quantile is minus the   pth quantile.

How to count n th percentile from normally distributed random variable? I have normally distr

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